Parking your bike in that there London.

Can I just chain my bike to a near by lamp post

Only if there isn't a sticker saying "don't park a bike here"

There will more than likely be some bike storage provision nearby, bring a shitty bike and three D locks if you want it to still be there at midnight.

I was in a secure car-park recently in central London, one with a roller door thingy, and it had bike-parking.

I think it was QPark, their web-site has some inf-- https://www.q-park.co.uk/en-gb/case-studies/bike-parking/

If you're just looking for a lamp post that OK for locking, or even proper bike racks, try this https://activethings.app/home

Can I just chain my bike to a nearby lamp post

Basically yes. But most London lamp posts are too fat to get a D-lock round near the bottom.
Be wary of general signposts - lots only have teeny little signs on the top that can be knocked off and your bike then lifted off the post.

Railings attached to buildings usually have "no bikes" signs on.

Park with other bikes; use a bike that's less attractive to thieves than the others and use lock(s) that are harder to break than the others. Basically make opportunists go for one of the other options, not yours!

From memory there's a bunch of bike racks at the top of Carnaby St, which is a minute or so's amble from the Palladium.

Westminster Borough Council was, for a long time, one of the most vile bike hating councils, but even there you can normally find a decent bike park to lock your bike to, looking at google maps there are a few opposite the Palladium. The council has been Labour since the last election, so hopefully things will improve

Westminster City Council 😉

noticed this place the other day, never used (normal on Brompton) and appears there are a few in central London. Apparently have 24/7 access which is better than some of the more commuter focused places that close from shows/pubs close.

https://www.spokesafe.com/chinatown